# Markov inequality

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

For a nonnegative random variable with finite expectation, its tail probability at a positive threshold is at most E[X] divided by that threshold.

mko-markov-inequality · theorem · v0.10.0



## Mathematical statement

Require X≥0 almost surely, a>0 and finite E[X]; then P(X≥a)≤E[X]/a. Complete finite rational laws allow exact tail and bound calculations. The bound is not the tail probability, signed variables cannot directly use the nonnegative theorem, and threshold0 is excluded.





## Representative formula (exact TeX)

M(X,a)





## Assumptions


- The complete normalized finite rational law and total variable-value table are supplied.
- Main theorem exercises require every table value nonnegative and the threshold positive.





## Symbol scope


- X — nonnegative random variable
- a — positive threshold





## Intuition, definition, and misconceptions

Nonnegativity, a positive threshold and finite expectation are explicit hypotheses.

The actual tail is a sum over the complete law.

Markov supplies an upper bound, not equality.

Do not apply the nonnegative theorem to a signed variable or call E[X]/a the actual probability.





## Adopted derivation guide

[Deriving a Markov bound pointwise](https://ocme.evemisslab.com/en/derivations/derivation-mko-markov-inequality-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-random-variable/) — X and its law are specified.
- [Expectation in finite models](https://ocme.evemisslab.com/en/mko/mko-expectation/) — The bound uses E[X].
- [Probability on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-probability/) — The event is a threshold tail.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Nonnegativity and the threshold comparison use the order.





## Verification boundary

Exact exercises use complete finite laws; the general measure version is stated, not proved by finite samples. The prototype pointwise check is stronger than almost-sure nonnegativity at null-mass entries; ambiguous negative-null cases need source review.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add Chebyshev inequality.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/markov-inequality.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-markov-inequality/index.md)